How to fold a disk into a cylinder while the edge are touching
Let's try this:


Create a cylinder (Fill Type > Ngon), rotate it so it faces the view, apply the rotation with a ctrl A.
In Edit mode, inset with i.
Inset again and alt M > Merge at Center.
Create some edge loops with ctrl R to add some topology.
Give your object a Simple Deform modifier, Bend mode, with a 360° Deform Angle.
Give it a Subdivision Surface modifier, press the Smooth Shading button on the Tools panel.
To give your object some thickness you can add a Solidify modifier, put it above your Subdivision Surface modifier, and choose the Thickness value that suits you.
To have a sharp edge make sure to have an edge loop close to the edge.